Thus there is a natural bijection from the set of all possible equivalence relations on X and the set of all partitions of X.

Thus an equivalence relation over X, a partition of X, and a projection whose domain is X, are three equivalent ways of specifying the same thing.

Thus it is meaningful to speak of a " presentation of an equivalence relation ," i. e., a presentation of the corresponding groupoid ;

Thus, S < sub > i </ sub > is a subset of an equivalence class of R < sub > L </ sub >.

Thus, the existence of a finite automaton recognizing L implies that the Myhill – Nerode relation has a finite number of equivalence classes, at most equal to the number of states of the automaton, and the existence of a finite number of equivalence classes implies the existence of an automaton with that many states.

Thus, it is a highly idiomatic translation, and as such falls on the extreme dynamic end of the dynamic and formal equivalence spectrum.

Thus the classical solutions to G Chern – Simons theory are the flat connections of principal G-bundles on M. Flat connections are determined entirely by holonomies around noncontractible cycles on the base M. More precisely, they are in one to one correspondence with equivalence classes of homomorphisms from the fundamental group of M to the gauge group G up to conjugation.

Thus, evaluators need to take into account two important concepts when administering data collection tools: lexical equivalence and conceptual equivalence ( ibid ).

Thus the pH will equal the pK < sub > a </ sub > which occurs half-way to the equivalence point.

Thus the relation of logical equivalence partitions all primary arithmetic expressions into two equivalence classes: those that simplify to the Cross, and those that simplify to the void.

Thus Q defines an inner product on dom T. Let H < sub > 1 </ sub > be the completion of dom T with respect to Q. H < sub > 1 </ sub > is an abstractly defined space ; for instance its elements can be represented as equivalence classes of Cauchy sequences of elements of dom T. It is not obvious that all elements in H < sub > 1 </ sub > can identified with elements of H. However, the following can be proved:

Thus, after flashing furniture, the equivalence of chair-chair is detected more rapidly than stove-stove.

Thus they left open the possibility that the Ape Index may be more significant when there is a greater degree of equivalence between the other traits under consideration.

Thus a novel might be translated with greater use of dynamic equivalence so that it may read well, while in diplomacy or in some business settings people may insist on formal equivalence because they believe that fidelity to the grammatical structure of the language equals greater accuracy.

Thus we can determine the equivalence ratio of the given mixture as,

Thus the relevant notion of topological equivalence is a considerable weakening of the naïve C < sup > 1 </ sup > conjugacy of vector fields.

Thus, the combined efficiency of the elements replaced by the two fiber plates ( with a combined efficiency of 0.25 ) is 0.043 or about six times less than that of the two fiber plates.

Thus if the universe is hot enough ( approximately 10 < sup > 15 </ sup > K, a temperature exceeded until shortly after the Big Bang ) then the electromagnetic force and weak force will merge into a combined electroweak force.

Thus in March 1979 Colonel Muammar Gaddafi, who kept absolute power ( until his overthrow in 2011 referred to as " Guide of the Revolution "), after ten years as combined Head of State and Head of government of the Libyan Jamahiriya (" state of the masses "), styled Chairman of the Revolutionary Command Council, formally transferred both qualities to the General secretaries of the General People's Congress ( comparable to a Speaker ) respectively to a Prime Minister, in political reality both were his creatures.

Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group ; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi ; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.

Thus there was no such thing as combined planning of the different services.

Thus explanations of a sort are provided for the astrological effects of the planets, based upon their combined effects of heating, cooling, moistening, and drying.

Thus, one had a theory which combined gravity, quantization and even the electromagnetic interaction, promising ingredients of a fundamental physical theory.

Thus the existing system for measuring distance, combined with a memory capacity to see where the target last was, is enough to measure speed.

Thus, it was Gibbs who first combined the first and second laws of thermodynamics by expressing the infinitesimal change in the energy a system in the form:

Thus Offa's Dyke is a combined structure and Car Dyke is a trench though it once had raised banks as well.

Thus the combined size of all the instructions needed to perform a particular task, the code density, was an important characteristic of any instruction set.

Thus, all three series are set in the same universe, giving them a combined length of 15 novels, and a total of about 1, 500, 000 words.

Thus, when meteoroids meet the Earth's atmosphere head-on ( which would only occur if the meteors were in a retrograde orbit ), the combined speed may reach about 44 miles per second ( 71 kilometers per second ).

Thus, all measured nakhlites give essentially identical CRE ages of approximately 11 Ma, which when combined with their possible identical formation ages indicates ejection of nakhlites into space from a single location on Mars by a single impact event.

Thus, the Greek god of interpretive communication was combined with the Egyptian god of wisdom as a patron of astrology and alchemy.

Thus, a legacy switch system can be easily " converted " to allow wavelength multiplexed transport over a fiber simply by judicious choice of transceiver wavelengths, combined with an inexpensive passive optical multiplexing device.

Thus, some gluten-free diets are combined with casein-free diets and referred to as a gluten-free, casein-free diet.

Thus negi and tofu, a strongly flavored ingredient mixed with a delicately flavored ingredient, are often combined.

Thus the Presbyterians of the north, who were mainly republican in sentiment, combined with a section of the Roman Catholics to form the United Irishmen organisation, promoting revolutionary ideas imported from France ; and a party prepared to welcome a French invasion soon came into existence.

Thus the relatively small amount of network color programming, combined with the high cost of color television sets, meant that as late as 1964 only 3. 1 percent of television households in the U. S. had a color set.

Thus Offa's Dyke is a combined structure and Car Dyke is a trench, though it once had raised banks as well.

Thus, Chiang had sent a large number of troops into positions in which they could neither defend properly, without the combined support of the army, nor defend with any real purpose, setting the stage for the last great struggle of the Warlord Era.

Thus it is important to remember that the favorable economic position of households in the top two quintiles is in some cases the result of combined income, rather than demand for a single worker.

( Thus, for example, Smith's justification for believing that the person who will get the job has ten coins in his pocket is his justified belief that Jones will get the job, combined with his justified belief that Jones has ten coins in his pocket.

Thus, to cite but one example, the Pax Britannica of the nineteenth century, whether with the British navy ruling the seas or with the City of London ruling world finance, was strictly national in motivation, however much other nations ( e.g., the United States ) may have incidentally benefited.

Thus, paradoxically, the beat writers resort to `` religious '' metaphors: they are in search of mana, the spiritual, the numinous, but not anything connected with formal religion.

Thus the transformation of Adam Smith's ideal entrepreneur into a mythological detective coincides closely with the decline of the real entrepreneur in economic life.

Thus, if corporations are not to run away with us, they must become quasi-governmental institutions, subject to public control and needs.

Thus with regard to the loss of tradition, in the change from order to disorder the metaphysics of change works itself out as a disruption of the individual soul, a change in which man continues as an objective ontological existent, but no longer as a man.

Thus, Margenau remarks: `` A large number of unrelated epicycles was needed to explain the observations, but otherwise the ( Ptolemaic ) system served well and with quantitative precision.

Thus he complains, with considerable justice, that the Tory writers have resorted to libel instead of answering his arguments.

Thus Burns's `` My love is like a red, red rose '' and Hopkins' `` The thunder-purple sea-beach, plumed purple of Thunder '' although clearly intelligible in content, hardly present ideas of the sort with which we are here concerned.

Thus, the Church was born and because of its intrinsic character was soon identified as a conservative institution, determined to resist the forces of change, to identify itself with the political rulers, and to maintain a kind of splendid isolation from the masses.

Before them stalked the beadle, proclaiming as he went, `` Thus the Council deals with those who break its laws -- adulterers, thieves, murderers, and lewd persons.

Thus was invented the single thread sewing machine, which Mr. Gibbs in partnership with Mr. Willcox decided to bring to Brown & Sharpe with the proposal that the small company undertake its manufacture.

Thus, when you have prepared your foundation and laid the floor, these can be trucked to the site and erected with a small crew of friends in a weekend.

Thus, direct comparisons can be drawn with free burning arcs which have been studied in detail during the past years and decades by numerous investigators ( Ref. 3 ).

Thus, the alignment of the `` dots '' and `` tips '', respectively, indicate individual variability of the 21 growth centers of each child with respect to the mean values for these boys and girls.

Thus cortico-fugal discharges induced by topical application of strychnine to a minute area in the neocortex summate with spikes present in the hypothalamus and cause increased convulsive discharges.

Thus D' and N' commute with any polynomial in T ; ;

Thus the unstressed it of it rarely snows here gets its significance from its use with snows: nothing can snow snow but `` it ''.

Thus in Mary wrote an account of the trip first strong stress on Mary marks Mary as the first in a series of people who wrote accounts of the trip, strong stress on wrote marks the writing as the first of a series of actions of Mary's concerned with an account of her trip ( about which she may later have made speeches, for example ), and strong stress on trip makes the trip the first of a series of subjects about which Mary wrote accounts.

Thus, the Commission acted with a sense of social responsibility within the area of its own convictions about the problem of government support to private education.

Thus he wired J. P. Lower and Sons of Denver: `` Have you any percussion hand grenades for throwing in a house or across a well loaded with balls or shrapnel shot??

Thus every part and plane of the picture keeps changing place in relative depth with every other part and plane ; ;

Thus, in the example cited above Fromm rests his whole case on the premise that the workers are being deprived unconsciously, unknowingly, of fulfillment, and then supports this with survey data reporting conscious, experienced frustrations.